Progress In Electromagnetics Research B, Vol. 13, 309–328, 2009
ANALYTICAL EXPRESSION OF THE MAGNETIC
FIELD CREATED BY TILE PERMANENT MAGNETS
TANGENTIALLY MAGNETIZED AND RADIAL CURRENTS IN MASSIVE DISKS
R. Ravaud and G. Lemarquand
Laboratoire d’Acoustique de l’Universite du Maine, UMR CNRS 6613
Avenue Olivier Messiaen, 72085 Le Mans, France
Abstract—In this paper, we present new expressions for calculating
the magnetic field produced by either tile permanent magnets
tangentially magnetized or by radial currents in massive disks. These
expressions are fully analytical, that is, we do not use any special
functions for calculating them. In addition, they are three-dimensional
and can be used for calculating the magnetic field for all regular
points in space. The expressions commonly used for calculating the
magnetic field produced by radial currents in massive disks are often
based on elliptic integrals or semi-analytical forms. We propose in
this paper an alternative analytical method that can also be used for
tile permanent magnets. Indeed, by using the analogy between the
coulombian model and the amperian current model, radial currents
in massive disks can be represented by using the fictitious magnetic
pole densities that are located on two faces of a tile permanent magnet
tangentially magnetized. The two representations are equivalent and
thus, the shape of magnetic field produced is the same for all points
in space, with a smaller value in the case of it is produced by radial
currents in massive disks. Such expressions can be used for realizing
easily parametric studies.
1. INTRODUCTION
The calculation of the magnetic field produced by radial currents in
massive disks has been studied by Babic, Akyel, Salon and Kincic [1, 2].
Indeed, they obtained semi-analytical forms of the magnetic field
Corresponding author: G. Lemarquand (guy.lemarquand@univ-lemans.fr).
310
Ravaud and Lemarquand
produced by radial current in massive disks by using the BiotSavart Law. We propose in this paper to use another approach
for calculating the three components of the magnetic field produced
by radial current in massive disks. Indeed, we use the Coulombian
model instead of the Biot-Savart Law. By using the analogy between
the coulombian model and the amperian current model, we can
show that radial currents in massive disks can be represented as
tile permanent magnets whose polarization is perfectly tangential.
In this case, the shape of the magnetic field produced by radial
current in massive disks is exactly the same as the one produced
by a tile permanent magnet tangentially magnetized. However, it
is noted that magnetic fields produced by radial currents are always
weaker than the ones produced by permanent magnets. Some previous
calculations of the magnetic field produced by tile or arc-shaped
permanent magnets were determined with the coulombian model [3–
6]. In addition, other authors have proposed three-dimensional semianalytical expressions for calculating the magnetic field produced by
thick annular conductors [7–9]. More generally, such semi-analytical
expressions can be employed for optimizing devices using cylindrical
permanent magnets [10–18]. The expressions obtained are often
based on one or two numerical integrations or elliptic integrals [19–
21]. Consequently, some authors have proposed the use of toroidal
functions for the computation of the external magnetic field produced
by cylindrical permanent magnets [22–24].
The knowledge of the magnetic field produced by arc-shaped
permanent magnets is always interesting for the design of actuators
or sensors [25–36].
This paper is composed in two parts.
The first part deals with the analytical calculation of the magnetic
field produced by a tile permanent magnet tangentially magnetized.
The three expressions of the magnetic components are determined by
using the coulombian model. Studying the magnetic field produced
by tile permanent magnets tangentially magnetized can be interesting
because such tile permanent magnets can be used for the design
of magnetic couplings. We have represented in Fig 1 an example
of structure using tile permanent magnets whose polarizations are
perfectly tangential but with alternate polarizations. In addition,
such tile permanent magnets can be used for the design of electrical
machines. Indeed, in a classical electrical machine, the radial field
created by the stator constituted by an assembly of tile permanent
magnets whose polarizations are often radial can also be optimized
by adding tile permanent magnets tangentially magnetized in this
structure. This way of assembling tile permanent magnets with several
Progress In Electromagnetics Research B, Vol. 13, 2009
H ( r,
311
,z)
u
0
ur
uz
Figure 1. Representation of an assembly of tile permanent magnets
tangentially magnetized with alternate polarizations.
magnetizations allow one to optimize more easily the shape of the radial
field produced. Moreover, the magnetic field created by assemblies of
tile permanent magnets radially and tangentially magnetized allow one
to obtain more intense magnetic fields inside such structures. This is
interesting for ironless structures with ferrofluid seals or for realizing
magnetic couplings. Furthermore, such tile permanent magnets are
used for realizing Halbach structures. These kinds of structures are
very interesting because they allow us to create an uniform and intense
field inside a ring of tile permanent magnets. Eventually, we think that
such tile permanent magnets will be more and more used for realizing
unconventional structures of stacked permanent magnets.
In the second part of this paper, we discuss the possibility of
using the expressions of the magnetic field components created by a
tile permanent magnet tangentially magnetized for the case of massive
disks with radial currents. By comparing the coulombian model and
the amperian current model, we show that the two calculations are
the same. Such an analogy is interesting because the expressions of
the magnetic field components produced by a tile permanent magnet
tangentially magnetized are fully analytical whereas the ones created
by radial currents in massive disks use elliptic integrals and one term
must be solved numerically [1]. Consequently, we show that it seems
to be possible to obtain a fully analytical expression of the magnetic
field produced by radial currents in massive disks.
312
Ravaud and Lemarquand
uz
z2
u
J
z1
2
1
0
r1
r2
ur
Figure 2. Representation of the geometry considered. The tile inner
radius is r1 , the tile outer radius is r2 , its angular width is θ2 − θ1 . Its
magnetic polarization is J and its height is z2 − z1 .
2. STUDY OF THE MAGNETIC FIELD PRODUCED BY
A TILE PERMANENT MAGNET TANGENTIALLY
MAGNETIZED
2.1. Notation and Geometry
The geometry considered and the related parameters are shown in
Fig. 2. The inner radius of the tile is r1 , its outer radius is r2 . Its
angular width equals θ2 − θ1 with θ2 > θ1 . It height is z2 − z1 with
z2 > z1 . Its magnetic polarization is denoted J and verifies for all
points in tile tile permanent magnet the following equation:
J = Juθ
(1)
This implies that the polarization of the tile permanent magnet
is perfectly tangential. The magnetic field produced by such a tile
permanent magnet can be determined by using the coulombian model.
Indeed, according to the Coulombian model, this tile permanent
magnet generates a magnetic field that can be deducted from its
magnetic potential Φ(r ) for all points in space. This magnetic
potential is given by:
⎞
⎛
· J
−∇
J · dSi
1 ⎝
+
dV ⎠
(2)
Φ(r ) =
4πμ0
(Si ) (V ) r − r̃ r − r̃ i
of
where Si represents one surface
the tile permanent magnet and V
its volume. In addition, r − r̃ represents the distance between a
Progress In Electromagnetics Research B, Vol. 13, 2009
313
n5
uz
n3
n2
u
n1
0
n4
n6
ur
Figure 3. Representation of the tile permanent magnet studied in
which the six vector normal units are defined.
point located on the tile permanent magnet and the observation point.
Therefore, there is a magnetic pole surface density σs∗ given by:
σs∗ = J · n
(3)
where n is the unit normal vector. In addition, there is a magnetic
pole volume density given by:
· J
σv∗ = −∇
(4)
Thus, according to the Coulombian model, a tile permanent magnet
can be represented by magnetic pole surface densities that are located
on the faces of the magnet and a magnetic pole volume density that is
located in the tile permanent magnet. Let us now define the normal
vector units of each face of the tile permanent magnet. For this
purpose, let consider the Fig. 3 in which the six vector normal units
∗ for representing the magnetic
are defined. We use the notation σs,i
pole surface density that appear on the face whose normal unit is ni .
Therefore, we obtain:
∗
= J · n1 = 0
σs,1
∗
= J · n2 = +J
σs,2
∗
= J · n3 = 0
σs,3
∗
σs,4
∗
σs,5
∗
σs,6
= J · n4 = −J
= J · n5 = 0
= J · n6 = 0
(5)
(6)
(7)
(8)
(9)
(10)
314
Ravaud and Lemarquand
The magnetic pole volume density σv∗ can be determined by using the
definition of the divergence in cylindrical coordinates. We deduct that:
· J = 0
σv∗ = −∇
(11)
Consequently, only two magnetic pole surface densities appear on the
tile permanent magnet tangentially magnetized. The first one −J is
located on the face whose angular abscissa equals θ1 . The second one
+J is located on the face whose angular abscissa equals θ2 . We show
a cross-section of the tile permanent magnet tangentially magnetized
with its magnetic pole surface densities in the (R, θ) plane in Fig. 4.
n2
n3
J
r
n1
u
2
r1
2
n4
1
uz
ur
Figure 4. Representation of the geometry considered. The tile inner
radius is r1 , the tile outer radius is r2 , its angular width is θ2 − θ1 . Its
its height is z2 − z1 .
magnetic polarization is J,
2.2. Three-dimensional Analytical Expression of the
Magnetic Potentail Created by the Tile Permanent Magnet
Tangentially Magnetized
The reciprocal distance, in cylindrical coordinates, between the source
point (r̃, θ̃, z̃) and the observation point (r, θ, z) is expressed as follows:
1
1
=
(12)
R r̃, θ̃, z̃ = 2
r − r̃ r 2 + r̃ 2 − 2rr̃ cos θ − θ̃ + (z − z̃)
where r̃ represents the position vector of the source point and r is the
position vector of the observation point. It is emphasized here that
this reciprocal distance is not regular for all points in space. There
is a discontinuity of this reciprocal distance when the magnetic field
Progress In Electromagnetics Research B, Vol. 13, 2009
315
is determined on the permanent magnet surfaces. Several approaches
can be used for solving this problem. One method has been proposed
by [3] with the use of the Lambda’s function. Another possibility
is to use the Cauchy principal value. By using the Gauss’ theorem,
the magnetic field value can be easily verified when the observation
point is placed on the permanent magnet surfaces. However, it can be
noted that the magnetic field is rarely determined on the permanent
magnets but rather outside of the tile permanent magnets and also
inside the permanent magnet for studying the demagnetizing field.
All the analytical expressions presented in this paper can be used for
studying the three magnetic field components in the near-field.
Therefore, the magnetic potential Φ(r, θ, z) produced by a tile
permanent magnet tangentially magnetized is given by:
r2 z2
J
R (r̃, θ2 , z̃) dr̃dz̃
Φ(r, θ, z) =
4πμ0 r1 z1
r2 z2
J
R (r̃, θ1 , z̃) dr̃dz̃
(13)
−
4πμ0 r1 z1
After integrating according to r̃ and z̃, the magnetic potential
Φ(r, θ, z) can be expressed as follows:
2
2
2
J (−1)(i+j+k) φ(ai,j , bi,j , ck )
Φ(r, θ, z) =
4πμ0
(14)
i=1 j=1 k=1
where
⎡
φ(ai,j , bi,j , ck ) = −zk − bi,j − a2i,j arctan ⎣ ⎤
ck
bi,j − a2i,j
⎦
⎡
⎤
ai,j ck
⎢
⎥
+ bi,j − a2i,j arctan ⎣ ⎦
bi,j − a2i,j bi,j + c2k
2
2
−ck log ai,j + bi,j +ck −ai,j log ck + bi,j +ck (15)
and
ai,j = ri − r cos(θ − θj )
(16)
bi,j = r + − 2rri cos(θ − θj )
ξk = z − zk
(17)
(18)
2
ri2
316
Ravaud and Lemarquand
Eq. (13) is in fact a special case in which the magnetic potential
has a fully analytical form. It is emphasized here that this is rarely
the case for tile permanent magnets. Though the components of the
magnetic field created by tile permanent magnets radially or axially
magnetized can often be expressed in terms of elliptic integrals of the
first, second and third kind, their magnetic scalar potential is often
more difficult to determine in a semi-analytical form. Moreover, it can
be easier to determine the magnetic field created by such tile permanent
magnets without using the vector or scalar potentials. We present in
the next section the expressions of the three components created by
the tile permanent magnet tangentially magnetized in fully analytical
expressions.
2.3. Magnetic Field Produced by the Tile Permanent
Magnet Tangentially Magnetized
A way of calculating the magnetic field produced by a tile permanent
magnet tangentially magnetized is to determine first the magnetic
potential Φ(r, θ, z) for all points in space. We keep the notation
ξk (Eq. (18)) for the equations of the three magnetic components
Hr (r, θ, z), Hθ (r, θ, z) and Hz (r, θ, z).
2.4. Derivation of the Three Components
The magnetic field created by the tile permanent magnet tangentially
magnetized can be deducted by calculating the following expression:
H = −∇Φ(r,
θ, z)
(19)
and the three components are given respectively by:
θ, z) · ur
Hr (r, θ, z) = −∇Φ(r,
θ, z) · uθ
Hθ (r, θ, z) = −∇Φ(r,
(20)
θ, z) · uz
Hz (r, θ, z) = −∇Φ(r,
(22)
(21)
The three expressions obtained are also given in a fully analytical form.
2.5. Radial Component Hr (r, θ, z)
The radial component Hr (r, θ, z) is expressed as follows:
Hr (r, θ, z) =
2
2
2
J (−1)(i+j+k) hr (i, j, k)
4πμ0
i=1 j=1 k=1
(23)
Progress In Electromagnetics Research B, Vol. 13, 2009
317
where
⎡
⎤
−ei,j + cos (θ − θj )2 fj
ξk fj + bi,j
⎦
tanh−1 ⎣ hr (i, j, k) =
2
fj (fj + bi,j )
fj bi,j + ξk
(24)
− cos (θ − θj )2 log ξk + bi,j + ξk2
r
r ei,j =
rri −1 + cos (θ − θj )2
(25)
ri
ri
We represent in Fig. 5 the radial component of the magnetic field
produced by a tile permanent magnet tangentially magnetized versus
the angle θ with the following parameters: r1 = 0.025 m, r2 = 0.028 m,
π
π
rad, θ2 = + 16
rad, z1 = 0 m, z2 = 0.003 m,
r = 0.024 m, θ1 = − 16
z = 0.0015 m, J = 1 T.
with
fj =
100000
Hr [A/m]
50000
0
− 50000
−100000
− 0.4
− 0.2
0
0.2
Angle [rad]
0.4
Figure 5. Representation of the radial component Hr versus the angle
theta θ with the following parameters: r1 = 0.025 m, r2 = 0.028 m,
π
π
rad, θ2 = + 16
rad, z1 = 0 m, z2 = 0.003 m,
r = 0.024 m, θ1 = − 16
z = 0.0015 m, J = 1 T.
2.6. Azimuthal Component Hθ (r, θ, z)
The azimuthal component Hθ (r, θ, z) of the magnetic field produced by
a tile permanent maget tangentially magnetized is expressed as follows:
2
2
2
J (−1)(i+j+k) hθ (i, j, k)
Hθ (r, θ, z) =
4πμ0
i=1 j=1 k=1
(26)
318
Ravaud and Lemarquand
40000
Htheta [A/m]
20000
0
− 20000
− 40000
− 0.4
− 0.2
0
0.2
Angle [rad]
0.4
Figure 6. Representation of the azimuthal component Hθ versus
the angle theta θ with the following parameters: r1 = 0.025 m,
π
π
rad, θ2 = + 16
rad, z1 = 0 m,
r2 = 0.028 m, r = 0.024 m, θ1 = − 16
z2 = 0.003 m, z = 0.0015 m, J = 1 T.
with
⎡
⎤
ξk bi,j + cj
cj + di,j
⎦
tanh−1 ⎣ 2
cj (bi,j + cj )
cj bi,j + ξk
+yj log ξk + bi,j + ξk2
hθ = −yj (27)
where
yj = sin(θ − θj )
di,j = r − rri cos(θ − θj )
cj = r 2 (cos (θ − θj ))2 − 1
2
(28)
(29)
(30)
We represent in Fig. 6 the azimuthal component of the magnetic field
produced by a tile permanent magnet tangentially magnetized versus
the angle θ with the following parameters: r1 = 0.025 m, r2 = 0.028 m,
π
π
rad, θ2 = + 16
rad, z1 = 0 m, z2 = 0.003 m,
r = 0.024 m, θ1 = − 16
z = 0.0015 m, J = 1 T.
Progress In Electromagnetics Research B, Vol. 13, 2009
319
15000
Hz [A/m]
10000
5000
0
− 5000
− 10000
− 15000
− 0.4
− 0.2
0
0.2
Angle [rad]
0.4
Figure 7. Representation of the axial component Hθ versus the angle
theta θ with the following parameters: r1 = 0.025 m, r2 = 0.028 m,
π
π
rad, θ2 = + 16
rad, z1 = 0 m, z2 = 0.003 m,
r = 0.024 m, θ1 = − 16
z = 0.001 m, J = 1 T.
2.7. Axial Component Hz (r, θ, z)
The axial component Hz (r, θ, z) of the magnetic field produced by a
tile permanent maget tangentially magnetized is expressed as follows:
2
2
2
J (−1)(i+j+k) hz (i, j, k)
Hz (r, θ, z) =
4πμ0
(31)
i=1 j=1 k=1
with
hz (i, j, k) = log ri − r cos(θ − θj ) + bi,j + ξk2
(32)
We represent in Fig. 7 the axial component of the magnetic field
produced by a tile permanent magnet tangentially magnetized versus
the angle θ with the following parameters: r1 = 0.025 m, r2 = 0.028 m,
π
π
rad, θ2 = + 16
rad, z1 = 0 m, z2 = 0.003 m,
r = 0.024 m, θ1 = − 16
z = 0.001 m, J = 1 T.
2.8. Application of the Expressions Determined for the
Study of Ironless Structures Using Tile Permanent Magnets
Tangentially Magnetized
All the expressions determined previously can be used easily for the
study of the magnetic field produced by ironless structures using
assemblies of tile permanent magnets tangentially magnetized. Indeed,
their computational cost is very low and they are very accurate. We
320
Ravaud and Lemarquand
have represented the three-dimensional representations of the radial
and azimuthal components of the magnetic field produced by an
assembly of tile permanent magnets tangentially magnetized as shown
in Figs. 8 and 9 respectively.
100
Hr [kA/ m]
0
− 100
0.02
0
− 0.02
r[m]
0
r[m]
00.02
02
− 0.02
Figure 8. Three-dimensional representation of the radial component
Hr versus r in polar coordinates with the following parameters: r1 =
0.025 m, r2 = 0.028 m, z1 = 0 m, z2 = 0.003 m, z = 0.001 m, J = 1 T
and 8 tile permanent magnets are used.
Htheta [ kA/ m]
50
0
− 50
0.02
0 r[m]
− 0.02
0
r[m]
00.02
02
− 0.02
Figure 9.
Three-dimensional representation of the azimuthal
component Hθ versus r in polar coordinates with the following
parameters: r1 = 0.025 m, r2 = 0.028 m, z1 = 0 m, z2 = 0.003 m,
z = 0.001 m, J = 1 T and 8 tile permanent magnets are used.
Progress In Electromagnetics Research B, Vol. 13, 2009
321
3. MAGNETIC FIELD PRODUCED BY RADIAL
CURRENT IN MASSIVE DISKS WITH THE
COULOMBIAN MODEL
We explain in this section how to calculate the magnetic field produced
by radial currents in massive disks by using the coulombian model.
Several authors have studied the magnetic field produced by such
structures [1, 9]. In this paper, we study the particular case in which
both surface current densities and a volume current density appear in
massive disks. Therefore, strictly speaking, our calculation does not
treat exactly the same configurations as [1] and [9] but the expressions
established in the first part of this paper can also be used for calculating
the three components of the magnetic field created by radial currents
in massive disks. In fact, there is an analogy between our configuration
studied in the first part of this paper and the configurations studied
in [1] and [9]. We present in the next section this analogy.
3.1. Basic Equations
It is well known that the magnetic field produced by either a permanent
magnet or a current distribution can be determined by using either the
scalar magnetic potential Φ(r, θ, z) or the vector magnetic potential
θ, z). We have shown in the first part that a tile permanent magnet
A(r,
tangentially magnetized generates a scalar magnetic potential for all
points in space. Let us now consider the magnetic vector potential
created by the tile permanent magnet tangentially magnetized studied
θ, z)
in the first part of this paper. This vector magnetic potential A(r,
that is expressed as follows:
⎛
⎞
J
J
∇
∧
dV
∧
dS
μ
μ
μ0
= 0⎝
+
0 ⎠
(33)
A
4π
(S) (V )
r−r
r − r Therefore, we can say that there is a surface current density given by:
J ∧ n
k∗ =
μ0
(34)
In addition, there is a volume current density given by:
∇ ∧ J
j∗ =
μ0
(35)
322
Ravaud and Lemarquand
3.2. Application to a Tile Permanent Magnet Tangentially
Magnetized
Let us now apply the previous relations the configuration presented in
Fig. 2. According to the Amperian current model, a tile permanent
magnet tangentially magnetized can be represented by using (34)
and (35). We find:
KS1 =
KS2 =
KS3 =
KS4 =
KS5 =
KS6 =
J
J
uθ ∧ (−ur ) = − uz
μ0
μ0
J
uθ ∧ uθ = 0
μ0
J
J
uθ ∧ (ur ) = + uz
μ0
μ0
J
uθ ∧ (−uθ ) = 0
μ0
J
J
uθ ∧ (uz ) = + ur
μ0
μ0
J
J
uθ ∧ (−uz ) = − ur
μ0
μ0
In addition, the volume current density is expressed as follows:
∇
1 1 ∂(rJ)
J
∗
uz = uz
∧ Juθ =
j =
μ0
μ0 r
∂r
r
(36)
(37)
(38)
(39)
(40)
(41)
(42)
We represent in Fig. 10 the surface current densities that are located
on four faces of the tile permanent magnet and in Fig. 11 the surface
current density that is located in the tile permanent magnet. An
interesting point to say is that current on the face S1 and the current
on the face S3 are not the same because S3 > S1 . This is why we can
say that a volume current density appears inside the tile permanent
magnet. This allows us to obtain a charge equilibrium of the magnet.
Therefore, the three magnetic components peuvent also be determined
by using the following relation:
∇
θ, z).ur
∧ A(r,
μ0
∇
θ, z).uθ
∧ A(r,
Hθ (r, θ, z) =
μ0
∇
θ, z).uz
∧ A(r,
Hz (r, θ, z) =
μ0
Hr (r, θ, z) =
(43)
(44)
(45)
Progress In Electromagnetics Research B, Vol. 13, 2009
323
uz
u
0
ur
Figure 10. Representation of the surface current densities that appear
on a tile permanent magne tangentially magnetized with the amperian
current model.
uz
u
0
ur
Figure 11. Representation of the volume current density that
appears inside a tile permanent magne tangentially magnetized with
the amperian current model.
In short, by assuming that there are radial currents in massive disks
that are characterized by both surface current densities k∗ and volume
current densities j ∗ , we can use directly the coulombian model by
J
n
and k∗ = J∧
These two
using the transformations j ∗ = ∇∧
μ0
μ0 .
transformations allow us to define the equivalent value of J that
correspond to a tile permanent magnet tangentially magnetized. The
interest of such an approach lies in the fact that the expressions of
the three magnetic components created by a tile permanent magnet
tangentially magnetized are fully analytical. Consequently, we can also
324
Ravaud and Lemarquand
find in a fully analytical form the three magnetic components created
by radial currents in massive disks.
3.3. Illustration: Comparison with the Biot-Savart Law
The relations determined in the previous section can be compared with
the well-known Biot-Savart Law. For this purpose, let us consider the
previous configuration shown in Figs. 10 and 11 in which both a volume
current density j ∗ and four surface current densities kS1 , kS3 , kS5 and
kS6 are considered. It is emphasized here that this configuration differs
from the one studied in [1] and [9]. The magnetic field produced by
such a current distribution is given by:
r 2 θ2 z 2 ∗
j dv ∧ (r − r )
1
H(r) =
3
4π r1 θ1 z1
r − r θ2 z 2 1
kS1 dS1 ∧ (r − r )
+
3
4π θ1 z1
r − r θ2 z 2 1
kS3 dS3 ∧ (r − r )
+
3
4π θ1 z1
r − r r 2 θ2 1
kS5 dS5 ∧ (r − r )
+
3
4π r1 θ1
r − r r 2 θ2 1
kS6 dS6 ∧ (r − r )
+
(46)
3
4π r1 θ1
r
−
r
For our numerical simulation, we take a surface current density kSi that equals 104 A/m2 and i is the surface i of the tile, r1 = 0.025 m,
m, z1 = 0 m, z2 = 0.003 m, θ1 = 0 rad, θ2 = π8 rad,
r2 = 0.028
4
and j∗ that equals 10 A/m3 , θ = π rad, z = 0 m. We use the
r
7
for the simulations and its calculation has
magnetic induction field B
been performed by numerical means. We represent in Fig. 12 the radial
component of the magnetic induction field versus the radial observation
point with the previous parameter values. Figure 12 confirms the
validity of the coulombian model for calculating the magnetic field
produced by radial currents in massive disks.
Progress In Electromagnetics Research B, Vol. 13, 2009
325
0.6
Br [ micro T ]
0.4
0.2
0
− 0.2
− 0.4
− 0.6
0.026
0.028
0.03
r [ m]
0.032
0.034
Figure 12. Representation of the radial component of the magnetic
determined with the coulombian model (thick line)
induction field B
and with the Biot-Savart Law (points). We take kSi = 104 A/m2
where i is the surface i of the tile, r1 = 0.025
m, r2 = 0.028 m, z1 = 0 m,
4
π
z2 = 0.003 m, θ1 = 0 rad, θ2 = 8 rad, j∗ = 10r A/m3 , θ = π7 rad,
z = 0 m.
4. CONCLUSION
We have presented new expressions for calculating the magnetic field
produced by either a tile permanent magnet tangentially magnetized,
or by radial currents in massive disks. All the expressions determined
in this paper are presented in a fully analytical form, that is,
without any special functions as elliptic integrals. Consequently, such
expressions have a very low computational cost and are accurate
because all the expressions have been determined without any
simplifying assumptions. The analogy between the coulombian model
and the amperian current model allows us to calculate with the same
expressions the magnetic field produced by tile permanent magnets
whose polarization is tangential or by radial currents in massive disks.
REFERENCES
1. Babic, S., C. Akyel, S. Salon, and S. Kincic, “New expressions for
calculating the magnetic field created by radial current in massive
disks,” IEEE Trans. Magn., Vol. 38, No. 2, 497–500, 2002.
2. Babic, S. and M. M. Gavrilovic, “New expression for calculating
magnetic fields due to current-carrying solid conductors,” IEEE
Trans. Magn., Vol. 33, No. 5, 4134–4136, 1997.
326
Ravaud and Lemarquand
3. Babic, S. and C. Akyel, “Improvement in the analytical calculation
of the magnetic field produced by permanent magnet rings,”
Progress In Electromagnetics Research C, Vol. 5, 71–82, 2008.
4. Ravaud, R., G. Lemarquand, V. Lemarquand, and C. Depollier,
“Analytical calculation of the magnetic field created by
permanent-magnet rings,” IEEE Trans. Magn., Vol. 44, No. 8,
1982–1989, 2008.
5. Ravaud, R., G. Lemarquand, V. Lemarquand, and C. Depollier,
“The three exact components of the magnetic field created
by a radially magnetized tile permanent magnet,” Progress In
Electromagnetics Research, PIER 88, 307–319, 2008.
6. Ravaud, R., G. Lemarquand, V. Lemarquand, and C. Depollier,
“Discussion about the analytical calculation of the magnetic field
created by permanent magnets,” Progress In Electromagnetics
Research B, Vol. 11, 281–297, 2009.
7. Azzerboni, B. and E. Cardelli, “Magnetic field evaluation for disk
conductors,” IEEE Trans. Magn., Vol. 29, No. 6, 2419–2421, 1993.
8. Azzerboni, B., E. Cardelli, M. Raugi, A. Tellini, and G. Tina,
“Magnetic field evaluation for thick annular conductors,” IEEE
Trans. Magn., Vol. 29, No. 3, 2090–2094, 1993.
9. Azzerboni, B. and G. Saraceno, “Three-dimensional calculation
of the magnetic field created by current-carrying massive disks,”
IEEE Trans. Magn., Vol. 34, No. 5, 2601–2604, 1998.
10. Furlani, E. P., Permanent Magnet and Electromechanical Devices:
Materials, Analysis and Applications, Academic Press, 2001.
11. Furlani, E. P., S. Reznik, and A. Kroll, “A three-dimensonal field
solution for radially polarized cylinders,” IEEE Trans. Magn.,
Vol. 31, No. 1, 844–851, 1995.
12. Furlani, E. P. and M. Knewston, “A three-dimensional field
solution for permanent-magnet axial-field motors,” IEEE Trans.
Magn., Vol. 33, No. 3, 2322–2325, 1997.
13. Furlani, E. P., “A two-dimensional analysis for the coupling of
magnetic gears,” IEEE Trans. Magn., Vol. 33, No. 3, 2317–2321,
1997.
14. Furlani, E. P., “Field analysis and optimization of ndfeb axial field
permanent magnet motors,” IEEE Trans. Magn., Vol. 33, No. 5,
3883–3885, 1997.
15. Mayergoyz, D. and E. P. Furlani, “The computation of
magnetic fields of permanent magnet cylinders used in the
electrophotographic process,” J. Appl. Phys., Vol. 73, No. 10,
5440–5442, 1993.
Progress In Electromagnetics Research B, Vol. 13, 2009
327
16. Elies, P. and G. Lemarquand, “Analytical optimization of the
torque of a permanent-magnet coaxial synchronous coupling,”
IEEE Trans. Magn., Vol. 34, No. 4, 2267–2273, 1998.
17. Lemarquand, V., J. F. Charpentier, and G. Lemarquand,
“Nonsinusoidal torque of permanent-magnet couplings,” IEEE
Trans. Magn., Vol. 35, No. 5, 4200–4205, 1999.
18. Lang, M., “Fast calculation method for the forces and stiffnesses
of permanent-magnet bearings,” 8th International Symposium on
Magnetic Bearing, 533–537, 2002.
19. Babic, S. and C. Akyel, “An improvement in the calculation of
the magetic field for an arbitrary geometry coil with rectangular
cross section,” International Journal of Numerical Modelling:
Electronic Networks, Devices and Fields, Vol. 18, 493–504,
November 2005.
20. Babic, S., C. Akyel, and S. Salon, “New procedures for calculating
the mutual inductance of the system: filamentary circular coilmassive circular solenoid,” IEEE Trans. Magn., Vol. 39, No. 3,
1131–1134, 2003.
21. Babic, S., C. Akyel, and M. M. Gavrilovic, “Calculation
improvement of 3d linear magnetostatic field based on fictitious
magnetic surface charge,” IEEE Trans. Magn., Vol. 36, No. 5,
3125–3127, 2000.
22. Selvaggi, J. P., S. Salon, O. M. Kwon, M. V. K. Chari, and
M. DeBortoli, “Computation of the external magnetic field, nearfield or far-field from a circular cylindrical magnetic source using
toroidal functions,” IEEE Trans. Magn., Vol. 43, No. 4, 1153–
1156, 2007.
23. Selvaggi, J. P., S. Salon, O. M. Kwon, and M. V. K. Chari,
“Computation of the three-dimensional magnetic field from
solid permanent-magnet bipolar cylinders by employing toroidal
harmonics,” IEEE Trans. Magn., Vol. 43, No. 10, 3833–3839, 2007.
24. Selvaggi, J. P., S. Salon, O. M. Kwon, and M. V. K. Chari,
“Calculating the external magnetic field from permanent magnets
in permanent-magnet motors — An alternative method,” IEEE
Trans. Magn., Vol. 40, No. 5, 3278–3285, 2004.
25. Ravaud, R., G. Lemarquand, V. Lemarquand, and C. Depollier,
“Ironless loudspeakers with ferrofluid seals,” Archives of Acoustics, Vol. 33, No. 4, 3–10, 2008.
26. Wang, J., G. W. Jewell, and D. Howe, “Design optimisation and
comparison of permanent magnet machines topologies,” IEE Proc.
Elect. Power Appl., Vol. 148, 456–464, 2001.
328
Ravaud and Lemarquand
27. Yonnet, J. P., “Permanent magnet bearings and couplings,” IEEE
Trans. Magn., Vol. 17, No. 1, 1169–1173, 1981.
28. Zhu, Z., G. W. Jewell, and D. Howe, “Design considerations
for permanent magnet polarised electromagnetically actuated
brakes,” IEEE Trans. Magn., Vol. 31, No. 6, 3743–3745, 1995.
29. Abele, M., J. Jensen, and H. Rusinek, “Generation of uniform
high fields with magnetized wedges,” IEEE Trans. Magn., Vol. 33,
No. 5, 3874–3876, 1997.
30. Baran, W. and M. Knorr, “Synchronous couplings with sm co5
magnets,” 2nd Int. Workshop on Rare-Earth Cobalt Permanent
Magnets and Their Applications, 140–151, Dayton, Ohio, USA,
1976.
31. Remy, M., G. Lemarquand, B. Castagnede, and G. Guyader,
“Ironless and leakage free voice-coil motor made of bonded
magnets,” IEEE Trans. Magn., Vol. 44, No. 11, 2008.
32. Berkouk, M., V. Lemarquand, and G. Lemarquand, “Analytical
calculation of ironless loudspeaker motors,” IEEE Trans. Magn.,
Vol. 37, No. 2, 1011–1014, 2001.
33. Blache, C. and G. Lemarquand, “High magnetic field gradients
in flux confining permanent magnet structures,” Journal of
Magnetism and Magnetic Materials, Vol. 104, 1111–1112, 1992.
34. Blache, C. and G. Lemarquand, “New structures for linear
displacement sensor with hight magnetic field gradient,” IEEE
Trans. Magn., Vol. 28, No. 5, 2196–2198, 1992.
35. Charpentier, J. F. and G. Lemarquand, “Optimization of
unconventional p.m. couplings,” IEEE Trans. Magn., Vol. 38,
No. 2, 1093–1096, 2002.
36. Lemarquand, G., “Ironless loudspeakers,” IEEE Trans. Magn.,
Vol. 43, No. 8, 3371–3374, 2007.